the Ship Victory on a Tuesday is a must. easy to find it sits alone in the middle of a car park. So no problem with parking. Alexanders is an open mike spot ,two/three numbers usualy very popular but not very folky.standard varies from excellent to naff.. just like a folk club. when you get there seek out Graham Bellinger or Mike Green for a spot or phone ahead. Alexanders is three mins fron the ship victory.so you could visit both on the same night.
Union vaults for the beer.
The Raven in the Cross Keys on lower bridge street has a varied program depends which sunday as whether it is a session , a singaround or a guest night
Check the Folk Orbit for info around chester as there is lots on everynight and only 15/20 mins drive to most.